2023 AMC 10B
All 25 problems, 75 minutes, scored 6 / 1.5 / 0. The answer key and the idea behind each problem are on this page too, folded away until you ask for them.
The problems
Show the answer key, topics and key insights
This gives away all 25 answers and the idea behind each one. Sit the paper first if you mean to.
The same key as a PDF 128 KB — a separate file from the paper, so one can be handed over without the other.
| # | Answer | Topic | Difficulty | Key insight |
|---|---|---|---|---|
| 1 | C | Ratios, Percents & Averages | Find the level every glass must end at first: the total juice split four ways. Then each full glass gives up the difference between full and that level. | |
| 2 | B | Ratios, Percents & Averages | A 20% discount means times 0.8 and a 7.5% tax means times 1.075. Together Carlos pays 0.86 of the original price, so divide 43 by 0.86. | |
| 3 | D | Circles | When a right triangle sits inside a circle with all three corners on it, its longest side is a diameter. So the diameters are 5 and 13. | |
| 4 | C | Sets, Estimation & Miscellaneous | Change both lengths into centimeters before multiplying: 6.5 mm is 0.65 cm and 25 m is 2500 cm. Then the area comes out directly in square centimeters. | |
| 5 | A | Linear Equations & Word Problems | Tripling every number triples the total, so the original total is 15. Adding 3 to every number adds 3 once for each number, and that extra is 30. | |
| 6 | E | Modular Arithmetic | Look only at odd or even. The terms go odd, odd, even, and that pattern repeats, so a term is even exactly when its position is a multiple of 3. | |
| 7 | B | Transformations & Symmetry | All eight corners lie on one circle around the center. Angle EAB is half of arc EB, and arc EB is 90 minus 20 = 70 degrees. | |
| 8 | A | Modular Arithmetic | Only the last digit of each base matters. Last digits of powers of 2 and of 3 repeat every 4 steps, so use the remainders of 2023 and 2022. | |
| 9 | B | Number Properties | Neighboring squares n squared and (n+1) squared always differ by 2n + 1. So count the values of n that make 2n + 1 at most 2023. | |
| 10 | C | Paths & Grids | Every possible rectangle covers one of the four middle-of-edge squares, so 4 guesses always work. Three guesses can touch at most 10 of the 12 rectangle positions. | |
| 11 | B | Diophantine Equations | Everything except the fifty-dollar bills comes in multiples of 20 or 100, so the fifties must come in pairs. Then list the choices by the number of hundreds. | |
| 12 | C | Polynomials | To the right of 10 every factor is positive. Moving left, P changes sign only when you pass a root with an odd power: 9, 7, 5, 3, 1. | |
| 13 | B | Absolute Value & Inequalities | The region looks the same in all four quadrants. In one quadrant it is a tilted square around (1, 1) with area 2, so the total is 4 times 2. | |
| 14 | C | Diophantine Equations | Add mn to both sides to get (m+n) squared on the left. Then (mn) squared plus mn must be a perfect square, so mn is 0 or -1. | |
| 15 | C | Number Properties | Pair each factorial with the next, like 3! times 4! = (3!) squared times 4. What remains is a square times 8!, and 8! needs 2 times 5 times 7. | |
| 16 | E | Basic Counting | Picking the digits decides the number, so count sets of digits. The digit 0 can end a downno but can never start an upno. | |
| 17 | D | Algebraic Manipulation | The diagonal squared is a squared plus b squared plus c squared. That equals (a+b+c) squared minus 2(ab+bc+ca), and the edge total and face total give both parts. | |
| 18 | E | GCD & LCM | Multiply by 210 to get c = 15a + 14b. Then 2 and 7 divide c exactly when they divide a, and 3 and 5 exactly when they divide b. | |
| 19 | B | Geometric Probability | All four directions are alike, so assume she hops north. She leaves when her height plus the hop passes 6: a small triangle in a 1 by 6 rectangle. | |
| 20 | A | Solid Geometry | The four corners where the semicircles meet are evenly spaced around the sphere's equator. Neighboring corners are 2 times root 2 apart, so each semicircle has radius root 2. | |
| 21 | E | Conditional Probability & States | Only odd or even matters for each bin. Track one question, 'do all three bins match?', ball by ball. The chance quickly settles at 1/4. | |
| 22 | B | Number Properties | Call the floor k. Then x = (k squared + 2)/3, and this x must really have floor k. Only k = 0, 1, 2, 3 pass that test. | |
| 23 | B | Sequences & Series | The real sum is 221 or 223. The number of terms divides twice the sum and is at most 14, since steps are at least 2. Only 13 fits. | |
| 24 | E | Coordinate Geometry | For each fixed w the points form a 2 by 1 rectangle. As w goes from 0 to 1, the rectangle slides along (-3, 4) and sweeps out a hexagon. | |
| 25 | B | Angles & Polygons | Folding a corner to the center creases halfway between them. So the new center-to-side distance is half the old center-to-corner distance: a golden-ratio shrink. |
2023 AMC 10B. Problems © Mathematical Association of America, American Mathematics Competitions (maa.org/amc), licensed CC BY-NC-SA 4.0. Typeset and annotated by AMC 10 Step by Step (amcstep.com) and shared under the same licence. Not endorsed by the MAA.